Multilevel polynomial regression analysis [MPRA; see Nestler et al. (2019)] is an adaptation of
polynomial regression analysis [PRA; see Edwards
& Parry (1993)] that can be used when data have a multilevel
structure (e.g., observations across multiple time points nested within
participants). In the current study, MPRA allows testing and
interpreting both the linear effects of vitality and learning
separately, as well as the effects of congruent vitality and learning
scores using data collected at four time points with a one-month time
lag between the assessments. In the following, we use the index m to
denote the Level 2 unit (\(m = 1…N\))
and the index \(i\) to denote the Level
1 unit (\(i = 1, 2, …, n_{m}\)). To
assess the existence of congruence effects, the following parameters are
estimated at the population level: the slope of the line of congruence
(LOC) (i.e., vitality = learning) as related to the outcome variable is
given by \(\alpha_{1m} = b_{1} +
b_{2}\). Here, \(b_{1}\) is the
unstandardized regression coefficient for the (latent) vitality variable
and \(b_2\) is the regression
coefficient for the (latent) learning variable. The curvature along the
line of perfect agreement is specified by \(\alpha_{1m} = b_{3} + b_{4} + b_{5}\).
Here, \(b_{3}\) is the regression
coefficient for (latent) vitality squared, b4 is the regression
coefficient for the cross-product of the (latent) vitality and the
(latent) learning variable, and b5 is the unstandardized regression
coefficient for latent learning squared. The slope of the line of
incongruence (LOIC) is defined as \(\alpha_{3m} = b_{1} - b_{2}\). The
curvature of the line of incongruent vitality and learning levels as
related to the outcome variable is specified as \(\alpha_{4m} = b_{3} - b_{4} + b_{5}\). Each
of the regression coefficients \(b_{1}\) to \(b_{5}\) includes a fixed effect and Level 2
residual terms. The fixed effects of the regression coefficients are
used to estimate the average response surface parameters \(\hat{\alpha}_{1}\) to \(\hat{\alpha}_{5}\) (Nestler et al., 2019).
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References
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The limits of the z-axis were restricted to reflect physical health values present among 90 percent of the participants (i.e., using the 0.05 and 0.95 quantiles as axis limits). The dots represent the raw data points. The x and y axes represent the person-mean centred vitality and learning scores, respectively.
The limits of the z-axis were restricted to reflect physical health values present among 90 percent of the participants (i.e., using the 0.05 and 0.95 quantiles as axis limits). The dots represent the raw data points. The x and y axes represent the person-mean centred vitality and learning scores, respectively.